Coat a large skillet or Dutch oven with vegetable oil. Cook the ground beef over medium heat, breaking it up into small pieces with a wooden spoon, until browned, 3 to 5 minutes. Stir in the onions and bell peppers and cook until the onions are translucent and just beginning to brown, 2 to 3 minutes.
Combine the ketchup, brown sugar, Dijon mustard, Worcestershire sauce, garlic powder, paprika, chili powder and black pepper in a small bowl. Add the sauce to the ground beef and mix well.
Lower the temperature and let the ground beef simmer for 20 minutes with the lid on. Taste and adjust seasoning. Stir in the butter and remove from heat. (See Cook's Note.)
To assemble, lay a slice of Cheddar on the bottom half of each doughnut. Spoon sloppy joe mixture over the cheese, then cover with the top half of each doughnut. Serve immediately.
Sloppy joes taste better the next day, so make a day in advance.